Work, Energy and Power

1.0Work

In physics, the word ‘work’ has a definite and precise meaning. Work is not done on an object unless the object is moved with the action of a force. The application of a force alone does not constitute work. In common usage, the word ‘work’ means any physical or mental exertion. But in physics, work has a distinctly different meaning. Let us consider the following situations :

(1) A student holds a heavy chair at arm’s length for several minutes.

(2) A student carries a bucket of water along a horizontal path while walking at constant velocity.

(3) A student applying force against a wall. 

(4) A student studying the whole day to prepare for examinations.

It might surprise you to know that in all the above situations, no work is done according to the definition of work in physics, even though effort is required in all cases.

Concept of Work

In physics, the word ‘work’ has a definite and precise meaning. Imagine that your car
(see figure) has run out of fuel and you have to push it down the road to the gas station. Let your friend push the car with a constant horizontal force. If the car does not move, no work is done. Suppose, he increases the magnitude of this force by pushing the car harder. If the car starts moving, he does a work on the car.

Work is not done on an object unless the object is moved with the action of a force. The application of a force alone does not constitute work. For example, when a student holds the chair in his hand, he exerts a force to support the chair. But, work is not done on the chair as the chair does not move.

Some more examples to understand the concept of work are given below :

(1) Push a box lying on a surface. The box moves through a distance. You exerted a force on the box and the box got displaced. In this situation work is done [see figure(a)]

(2) A man pushes a trolley and the trolley moves through a distance. A man has exerted a force on the trolley and it is displaced. Therefore, work is done [see figure(b)].

(3) A person lifts a cat through a certain height. To do this he applies a force. The cat rises up. There is a force applied on the cat and the cat has moved. Hence, work is done [see figure(c)].

Two important conditions that must be satisfied for work to be done are:

• A force should act on an object 
• The object must be displaced.
• Direction of force must not be perpendicular to the displacement.

If any one of the above conditions does not exist, work is not done. This is the concept of work that we use in science.

2.0Mathematical Definition of Work

(1) A constant force is applied in the direction of the displacement of an object: Let a constant force, F acts on an object. Let the object be displaced through a distance, s in the direction of the force (see figure). Let W be the work done.

 Here, we define work to be equal to ‘the product of the force and displacement’.

Work done = force × displacement

$W=F\times s$

(2) A constant force is applied at a certain angle with the direction of the displacement of an object : When the force on an object and the object’s displacement are in different directions, the work done on the object is given by, 

$W=F\times s\times cos$

where, the angle between the force and the direction of the displacement is θ (see figure).

Here, we define work to be equal to ‘the force multiplied by the displacement multiplied by the cosine of the angle between them’.

Work is a scalar quantity, it has only magnitude and no direction.

SI unit of work:  Joule        

1 Joule = 1 newton × 1 meter

or 1 J = 1 N m = 1 kg m² s^–2

C.G.S unit of work : Erg.

1 erg = 1 dyne cm = 1 g cm² s^–2

1 Joule = 10⁷ ergs

Definition of 1 joule:

1 J is the amount of work done on an object when a force of 1 N displaces it by 1 m along the line of action of the force.

The unit of work ‘Joule’ is named after the British physicist James Prescott Joule (1818–1889). Joule made major contributions to the understanding of energy, heat and electricity.

Some important points related to work

If θ = 0°, then cos 0° = 1 and W = F × s. 

If θ =90°, then, W = 0 because cos 90°= 0. So, no work is done on a bucket being carried by a girl walking horizontally (see figure). The upward force exerted by the girl to support the bucket is perpendicular to the displacement of the bucket, which results in no work done on the bucket.

If, θ = 180°, then cos 180° = –1 and W = – F × s.

Concept of Negative and Positive Work

The work done by a force can be either positive or negative.

(1) Whenever angle (θ) between the force and the displacement is acute, i.e., 0° < θ < 90°, the work done is positive. Also, when angle (θ) between the force and displacement is zero, i.e., force and displacement are in same direction, the work done is positive.

(2) Whenever angle (θ) between the force and the displacement is obtuse, i.e., 90°< θ <180°, the work done is negative. Also, when angle (θ) between the force and displacement is 180°, i.e., force and displacement are in opposite direction, the work done is negative.

Whenever force is in the direction of motion, velocity of the object increases and the work done is positive. Whenever force opposes motion, velocity of the object decreases and the work done is negative.

If many constant forces are acting on an object, you can find the net work done on the object by first finding the net force on the object.

Work done by the centripetal force is always zero because it is always perpendicular to the displacement. For example, if an electron moves around a nucleus in a circular path due to centripetal force provided by the electric force between them, the work done by this force is zero.

Work done by applied force against gravity

If an object is lifted up to a certain height (see figure), definitely, a work is done by the applied force. The applied force must be equal to the weight (mg) of the object. This work done is given by,

W = F × s = (mg) × h   

Where, m = mass of object ; g = acceleration due to gravity ; 

h = height

or W=mgh

When a body is lifted up, the work done by the applied force is positive while work done by the gravity is negative. Similarly, when a book is put down from a certain height, the work done by the applied force is negative while work done by the gravity is positive.

3.0Energy

Energy is defined as the internal capacity for doing work.

When we say that a body has energy it mean that it can do work.

● Energy is a scalar quantity

● Dimensions : [M¹L²T^–2]

● SI UNIT : joule

● 1 cal = 4.2 joule

Without light that comes to us from the Sun, life on Earth would not exist. With the light energy, plants can grow and the oceans and atmosphere can maintain temperature ranges that support life. 

Although energy is difficult to define comprehensively, a simple definition is that energy is the capacity to do work.  Thus, when you think of energy, think of what work is involved. Let us take some examples :

(i) Energy must be supplied to a car’s engine in order for the engine to do work in moving the car. In this case, the energy may come from burning petrol.

(ii) When a fast moving cricket ball hits a stationary wicket, the wicket is thrown away. Thus, work is done by the energy in the moving ball on the wicket. 

(iii) An object placed at a certain height has the capability to do work. If it is allowed to fall, it will move downward i.e., a work will be done in this case.

(iv) When a raised hammer falls on a nail placed on a piece of wood, it drives the nail into the wood. Thus, energy of hammer does a work on nail.

(v) When a balloon is filled with air and we press it we notice a change in its shape. That is, we have done a work on balloon to change its shape. As long as we press it gently, it can come back to its original shape when the force is withdrawn. This means, balloon acquires energy due to which it regains its original shape.

(vi) If we press the balloon hard, it can even explode producing a blasting sound. Again, it acquires enough energy that does work to blast the balloon.

In all the above examples, the objects acquire the capability of doing work which is called energy.

SI unit of energy : Since energy is the capacity to do work, its unit is same as that of work, that is, joule (J). 1 J is the energy required to do 1 joule of work. Sometimes a larger unit of energy called kilo joule (kJ) is used, 1 kJ = 1000 J.

4.0Forms of Energy

Different Forms of Energy

Mechanical energy, electrical energy, optical (light) energy, acoustical (sound), molecular, atomic and nuclear energy. 

These forms of energy can change from one form to the other.

Examples:

$Electricalenergy\stackrel{Motor}{\to }Mechanicalenergy$

$Mechanicalenergy\stackrel{Generator}{\to }Electricalenergy$

$Lightenergy\stackrel{Photocell}{\to }Electricalenergy$

$Electricalenergy\stackrel{Heater}{\to }Heatenergy$

$Electricalenergy\stackrel{Radio/Speaker}{\to }Soundenergy$

$Nuclearenergy\stackrel{NuclearReactor}{\to }Electricalenergy$

$Chemicalenergy\stackrel{cell}{\to }Electricalenergy$

$Electricalenergy\stackrel{RechargeableCell}{\to }Chemicalenergy$

Mechanical waves require a material medium to travel. Sound is also a mechanical wave thus, it requires a material medium like air, water, steel, etc. for its propagation. It cannot travel through a vacuum. 

The world we live in provides energy in many different forms. The various forms include potential energy, kinetic energy, heat energy, chemical energy, electrical energy and light energy.

Mechanical energy

The capacity to do mechanical work is called mechanical energy. Mechanical energy can be of two types :

(1) Kinetic energy

(2) Potential energy

The sum of the gravitational potential energy and the kinetic energy is called mechanical energy.

Mass Energy Relation 

According to Einstein mass energy equivalence principle mass and energy are inter convertible i.e. they can be changed into each other.

Equivalent energy corresponding to mass m is E = mc²

where, m : mass of the particle

c : speed of light

Kinetic Energy

Kinetic energy is the energy associated with an object in motion.

Derivation of Kinetic Energy

Let a constant force F acts on a ball of mass m with an initial velocity u. The displacement of the ball be ‘s’, time taken to displace it be ‘t’, its final velocity be ‘v’ and acceleration produced in it be ‘a’ .

Now, work done, W = F × s

or W = (ma) × s ..... (1) (F = m × a)

Now, from third equation of motion, we have, v² = u² + 2as

or $a=\frac{v^2-u^2}{2s}$ ..... (2)

From (1) and (2), we get, 

$W=m\times \frac{v^2-u^2}{2s}\times s$

$W=\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2$ ..... (3)

It is clear from equation 3, the work done is equal to the change in the kinetic energy of an object. Equation 3 is called Work-Energy Theorem.

If u = 0, the work done will be

$W=\frac{1}{2}m{v}^2-\frac{1}{2}m(0{)}^2=\frac{1}{2}m{v}^2$

Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is

$E_k=\frac{1}{2}m{v}^2$

If the speed of an object is doubled, its kinetic energy becomes four times the initial value. This is because kinetic energy is directly proportional to the square of the speed of the object. If the mass of an object in motion is doubled, then its kinetic energy is also doubled, as kinetic energy is directly proportional to the mass of the object.

Work done by the frictional force is always negative because it decreases the kinetic energy of an object. Also, frictional force is always opposite to the direction of displacement.

The more the kinetic energy an object possesses, more the work it will do.

This statement can be concluded with the following activity.

1. Take a heavy ball. Drop it on a thick bed of sand. A wet bed of sand would be better. Drop the ball on the sand bed from a height of about 25 cm. The ball creates a depression. Repeat this activity from heights of 50 cm, 1 m and 1.5 m. Ensure that all the depressions are distinctly visible. Mark the depressions to indicate the height from which the ball was dropped.

2. You will find that more the height from which the ball is dropped, more will be depth of the depression formed by it in the wet bed of sand. This is because more the height from which the ball is dropped, more will be the speed with which it strikes the wet bed. This means more will be its kinetic energy and it will do more work i.e., will penetrate to a greater depth. 

A moving object can do work.

This statement can be concluded with the following activity.

1. Set up the apparatus as shown in figure. Place a wooden block of known mass in front of the trolley at a convenient fixed distance. Place a known mass on the pan so that the trolley starts moving.

2. The trolley moves forward and hits the wooden block. Fix a stop on the table in such a manner that the trolley stops after hitting the block. The block gets displaced. Note down the displacement of the block. This means work is done on the block by the trolley as the block has gained energy. 

3. The pan along with mass falls due to gravity. Since, trolley is attached to the pan, it also starts moving i.e., it gains kinetic energy. Now, when the moving trolley hits the wooden block, a portion of kinetic energy of the trolley is transferred to wooden block. Due to this the block moves and gets displaced to a certain distance. 

4. Repeat this activity by increasing the mass on the pan. More the mass of the pan, more will be the kinetic energy gained by trolley and hence, more will be the kinetic energy gained by the wooden block. Thus, more will be its displacement. We can conclude that ‘a moving object can do work’. An object moving faster can do more work than an identical object moving relatively slow.

5.0Potential Energy

The energy possessed by an object due to its position or configuration is called ‘potential energy’.

Consider the balanced smooth rock shown in figure. As long as the rock remains balanced, it has no kinetic energy. If it becomes unbalanced, it will fall vertically to the ground and will gain kinetic energy as it falls. The origin of this kinetic energy is potential energy present in the rock. Thus, potential energy is stored energy.

Potential energy is associated with an object that has the potential to move because of its position or configuration.

Energy is present in the above example, but it is not kinetic energy because there is no motion. It is potential energy.

Gravitational Potential Energy

The energy associated with an object due to the object’s position relative to a gravitational source is called gravitational potential energy.

Gravitational potential energy is energy due to an object’s position in a gravitational field. Imagine an egg falling off a table. As it falls, it gains kinetic energy. But, where does the egg’s kinetic energy come from? It comes from the gravitational potential energy that is associated with the egg’s initial position on the table relative to the floor. 

Derivation of potential energy 

An object increases its energy when raised through a height. This is because work is done on it against gravity while it is being raised.

The gravitational potential energy of an object at a point above the ground is defined as ‘the work done in raising it from the ground to that point against gravity’.

Let us consider an object of mass, m which is raised through a height, h from the ground (see figure). A force equal to the weight (mg) of the object is required to do this. The object gains energy equal to the work done on it. 

Work done on the object, 

W = force × displacement = mg × h = mgh

This work done on the object is the energy gained by the object. This is the potential energy (E_P) of the object. That is,

$E_p=W=mgh$

The work done against gravity depends on the difference in vertical heights of the initial and final positions of the object, and not on the path along which the object is moved. Figure shows a case where a block is raised from position A to B by taking two different paths. Let the height AB = h. In both the situations the work done on the object is mgh.

The potential energy of an object at a height depends on the ground level or the zero level you choose. An object in a given position can have a certain potential energy with respect to one level and a different value of potential energy with respect to another level.

Gravitational potential energy depends on height from an arbitrary zero level

It is important to note that the height, h, is measured from an arbitrary zero level. In the example of the egg, if the floor is the zero level, then h is the height of the table, and mgh is the gravitational potential energy relative to the floor. Alternatively, if the table is the zero level, then h is zero. Thus, the potential energy associated with the egg relative to the table is zero. 

Let us take another example : 

Suppose you drop a volleyball from a second-floor roof and it lands on the first-floor roof of an adjacent building (see figure). If the height is measured from the ground, the gravitational potential energy is not zero because the ball is still above the ground. But if the height is measured from the first floor roof, the potential energy is zero when the ball lands on the roof. 

If B is the zero level, then all the gravitational potential energy is converted to kinetic energy as the ball falls from A to B. If C is the zero level, then only part of the total gravitational potential energy is converted to kinetic energy during the fall from A to B.

Elastic Potential Energy

Imagine you are playing with a spring on a tabletop. You push a block into the spring, compressing the spring, and then release the block. The block slides across the tabletop. The kinetic energy of the block came from the stored energy in the compressed spring (see figure). This potential energy is called elastic potential energy.

Elastic potential energy is stored in any compressed or stretched object, such as a spring or the stretched strings of a tennis racket or guitar.

The length of a spring when no external forces are acting on it is called the relaxed length of the spring. When an external force compresses or stretches the spring, elastic potential energy is stored in the spring. The amount of energy depends on the distance the spring is compressed or stretched from its relaxed length.

Any spring that has been stretched or compressed has stored elastic potential energy. This means that the spring is able to do work on another object by exerting a force over some distance as the spring regains its original length. The energy stored in a spring is also called ‘strain potential energy’. The energy available for use when a deformed elastic object returns to its original configuration is called ‘elastic potential energy’.

Energy appears in many forms, such as heat, motion, height, pressure, electricity, and chemical bonds between atoms.

Energy Transformations

Energy can be converted from one form to another form in different systems, machines or devices. Systems change as energy flows from one part of the system to another. Parts of the system may speed up, slow down, get warmer or colder, etc. Each change transfers energy or transforms energy from one form to another. For example, friction transforms energy of motion to energy of heat. A bow and arrow transform potential energy in a stretched bow into energy of motion (i.e., kinetic energy) of an arrow. 

Law of Conservation of Energy

Energy can never be created or destroyed, just converted from one form into another. This is called the law of conservation of energy. The law of conservation of energy is one of the most important laws in physics. It applies to all forms of energy.Energy has to come from somewhere

The law of conservation of energy tells us that energy cannot be created from nothing. If energy increases somewhere, it must decrease somewhere else. The key to understanding how systems change is to trace the flow of energy. Once we know how energy flows and transforms, we have a good understanding of how a system works. For example, when we use energy to drive a car, that energy comes from chemical energy stored in petrol. As we use the energy, the amount left in the form of petrol decreases.

Conservation of Mechanical Energy

The mechanical energy i.e., the sum of potential and kinetic energies is constant in the absence of any frictional forces. This means that if you calculate the mechanical energy (E_m) at any two randomly chosen times, the answers must be equal. Let us take an example of free fall (here, the effect of air resistance on the motion of the object is ignored) :

Let us consider an object of mass ‘m’ at a certain height ‘h’ (see figure). Let it is dropped from this height from point A i.e., v_A = 0.

Mechanical energy at A, E_A = K.E. at A + P.E. at A = $\frac{1}{2}m{v}_A^2+mg{h}_A$

or E_A = (0)² + mgh = mgh

$E_A=\frac{1}{2}m(0{)}^2+mgh=mgh$ ..... (1) 

Let after a certain time ‘t’, it reaches point B after covering a distance ‘x’.

Now, from third equation of motion, we have,

$v_B^2=v_A^2+2gx=(0{)}^2+2gx$

$v_B^2=2gx$ .... (2)

Mechanical energy at B,

$E_B=\text{K.E. at B}+\text{P.E. at B}=\frac{1}{2}m{v}_B^2+mg{h}_B$

$E_B=\frac{1}{2}m(2gx)+mg(h-x)$

= mgx + mgh – mgx = mgh ... (3)

Finally, the ball reaches the ground (at point C) after covering a distance h. Again, from third equation of motion, we have,

$v_C^2=v_A^2+2gh=(0{)}^2+2gh$

$v_C^2=2gh$...... (4)

Mechanical energy at C,

$E_C=\text{K.E. at C}+\text{P.E. at C}=\frac{1}{2}m{v}_C^2+mg{h}_C$

$E_C=\frac{1}{2}m(2gh)+mg(0)=mgh$ ...... (5)

From eqs. (1), (3) and (5), we get,  E_A = E_B = E_C

This means total mechanical energy is conserved during the free fall of an object.

That is, $\frac{1}{2}m{v}^2+mgh=constant$

During the free fall of the object, the decrease in potential energy during a certain time interval in its path, appears as an equal amount of increase in kinetic energy.

6.0Conservative and Non-Conservative Force

A force is said to be conservative if work done by the force on a particle moving between two points does not depends on the path taken by the particle.

Ex. (i) Gravitational force, not only due to earth but in its general form as given by the universal law of gravitation is a conservative force.

Note: This diagram shows that gravitational force is always conserved.

In all 3 cases work done by gravity  

W = mgh

(ii) Elastic force in a stretched or compressed spring is a conservative force.

(iii) Electrostatic force between two stationary electric charges is a conservative force.

(iv) Magnetic force between two magnetic poles is a conservative force.

Non Conservative Force 

 A force is said to be non-conservative, if work done by or against the force in moving a body from one position to another, depends on the path.

Difference Between Conservative and Non-conservative Forces

Example of Conservative Force and Non-conservative Force

7.0Power

The engine in an old school bus could, over a long period of time, do as much work as jet engines do when a jet takes off. However, the school bus engine could not begin to do work fast enough to make a jet lift off. In this and many other applications, the rate at which work is done is more significant than the amount of work done. 

A lift needs a powerful motor to raise the car when it has a full load of people. The motor does many thousands of joules of work each second.

Power is the rate at which work is done. Power can also be defined as the rate at which energy is transferred.                                                                                                       

Power = $\frac{Workdone}{Time}$

SI unit of power : Watt (W).1 Watt = 1 joule/second or 1 W = 1 J s^–1 

Definition of 1 watt : If 1 joule work is done per second by a device or a machine, then the power of that device or machine is 1 watt.

The unit ‘Watt’ was named in honour of the Scottish engineer, James Watt, who made such great improvements in the steam engine that it increased the Industrial Revolution.

James Watt did experiments with strong horses and determined that they could lift 550 pounds a distance of one foot in 1 s. He called this amount of power ‘one horsepower’ (hp). Converting to SI units, 1 hp = 746 W = 0.746 kW

Commercial Unit of Energy

The unit joule is an extremely small unit. It is inconvenient to express large quantities of energy in terms of joule. We use a bigger unit of energy called kilowatt hour (kWh). It is called commercial unit of energy.

Definition of 1 kWh

If a machine or a device of power 1 kW or 1000 W is used continuously for one hour, it will consume 1 kWh of energy. Thus, 1 kWh is the energy used in one hour at the rate of 1000 W (or 1 kW).

1 kWh = 1 kW × 1 h = 1000 W × 3600 s = 3600000 J or 1 kWh = 3.6 × 10⁶ J.

Electric power plants don’t make electrical energy. Energy cannot be created. What power plants do is convert other forms of energy (chemical, solar, nuclear) into electrical energy. When someone asks you to turn out the lights to conserve energy, he is asking you to use less electrical energy. If people used less electrical energy, power plants would burn less oil, gas, or other fuels.

The energy used in households, industries and commercial establishments is usually expressed in kilowatt hour. For example, electrical energy used during a month is expressed in terms of ‘units’. Here, 1 ‘unit’ means 1 kilowatt hour.

Solved Examples

1. A porter lifts a luggage of 15 kg from the ground and puts it on his head 1.5 m above the ground. Calculate the work done by him on the luggage (Take, g = 10 m/s²).

Solution

Given, mass of luggage, m = 15 kg ; height, h = 1.5 m ; g = 10 m/s²; W=?

Work done,  = mgh = 15 × 10 × 1.5 m = 225 J

2.  An artificial satellite is moving around the Earth in a circular path under the influence of centripetal force provided by the gravitational force between them. What is the work done by this centripetal force?

Explanation

Centripetal force (F) is always perpendicular to the displacement (s) of the particle moving along a circular path. That is, the angle (θ) between them is. 

Now, work done, W = F s cos θ = F s cos = 0 [  $cos90^{\circ }=0$ ]

Thus, work done by this centripetal force is zero.

3. A force of 5 N is acting on an object. The object is displaced through 2 m in the direction of the force (see figure). If the force acts on the object all through the displacement, then what is the work done on the object?

Solution

Given, force, F = 5 N; displacement, s = 2 m; W = ?

Then, the work done, W = F × s = 5 × 2 = 10 J.

4. An object of mass 15 kg is moving with a uniform velocity of 4ms^–1. What is the kinetic energy possessed by the object?

Solution

Given, mass of the object, m = 15 kg ; velocity of the object, v = 4 ms^–1 ; kinetic energy, E_k = ?

$E_k=\frac{1}{2}m{v}^2=\frac{1}{2}\times 15\times (4{)}^2=120J$

5. Find the energy possessed by an object of mass 10 kg when it is at a height of 6 m above the ground. Given, g = 9.8ms^–2.

Solution

Given, mass of the object, m = 10 kg ;  displacement (height), h = 6 m ; acceleration due to gravity, g = 9.8 m s^–2.

Potential energy = mgh = 10 kg ×  9.8 m s^–2 ×  6 m = 588 J.

6. A rod of mass 'm' and length '' is placed in horizontal position. Find work done by internal force against gravity to take it from horizontal position to vertical position. 

Solution

Displacement between position of centre of mass of rod is _\frac{l}{2} (In parallel direction of gravity or external force.)

$W=mg\frac{l}{2}$